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November 2009 Optimal rates for plug-in estimators of density level sets
Philippe Rigollet, Régis Vert
Bernoulli 15(4): 1154-1178 (November 2009). DOI: 10.3150/09-BEJ184


In the context of density level set estimation, we study the convergence of general plug-in methods under two main assumptions on the density for a given level $λ$. More precisely, it is assumed that the density (i) is smooth in a neighborhood of $λ$ and (ii) has $γ$-exponent at level $λ$. Condition (i) ensures that the density can be estimated at a standard nonparametric rate and condition (ii) is similar to Tsybakov’s margin assumption which is stated for the classification framework. Under these assumptions, we derive optimal rates of convergence for plug-in estimators. Explicit convergence rates are given for plug-in estimators based on kernel density estimators when the underlying measure is the Lebesgue measure. Lower bounds proving optimality of the rates in a minimax sense when the density is Hölder smooth are also provided.


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Philippe Rigollet. Régis Vert. "Optimal rates for plug-in estimators of density level sets." Bernoulli 15 (4) 1154 - 1178, November 2009.


Published: November 2009
First available in Project Euclid: 8 January 2010

zbMATH: 1200.62034
MathSciNet: MR2597587
Digital Object Identifier: 10.3150/09-BEJ184

Keywords: density level sets , kernel density estimators , minimax lower bounds , plug-in estimators , rates of convergence

Rights: Copyright © 2009 Bernoulli Society for Mathematical Statistics and Probability


Vol.15 • No. 4 • November 2009
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